A topo on topos Alain Connes
Abstract : Alain Connes presents the intellectual approach that led Alexandre Grothendieck, from an “an- noying” writing that he had to do for Bourbaki on homological algebra, to discover and perfect the concept of topos and he tries to explain in what sense this notion has a considerable scope, thanks in particular to the nuances it introduces between the true and false (Seminar organizer : Frédéric Jaëck (ENS), Trans- cription : Denise Vella-Chemla)
So I hope that I will stay in the spirit of the seminar and I think that the spirit of the seminar is Grothendieck, above all. So what I’m going to do is trying to get as far back as possible Grothendieck’s thought and trying to explain precisely, as I said in my abstract, the journey that brought him to the topos, and above all, I will try to give you an illuminating metaphor for what is a topos, and explain to you what is extraordinary about this discovery, in the sense, especially for philosophers, in the sense that it introduces considerable nuances into the notion of truth. I will try to explain this by an example, because nothing like a good example to explain. There will be one of my slides called “at two steps from the truth” and I’m really going to give you an example of a topos which allows us to say that we are for example 10 steps from the truth, or that we are 15 steps from the truth, etc. So listen carefully. I’ll make you hear Grothendieck, Grothendieck’s voice, because Grothendieck did 100 hours of lectures in Buffalo in 1973, and in those 100 hours of lectures, there are things that interest us. Of course I will not make you listen to him for a long time but I will make you listen to a point in time, where he explains what a sheaf is to people who don’t know this at all. And he explains how he will do his course on topos. We will see, there will also be an even more funny interlude at some point, we will not hear Grothendieck’s voice but we will hear Yves Montand’s voice. You will see, finally, you will hear all that.
So the first image I show you is an image that I owe to Charles Alunni who sent me an email one day telling me that he would have liked to have Grothendieck’s second thesis. But at the time, when we were doing a thesis, when I did my thesis for example, there was always a second thesis. This second thesis was not written. It was a second thesis that we had to defend in front of the jury. And we had a subject that was given to us. What is quite extraordinary is that in Grothendieck’s case, he did his thesis on nuclear spaces, on topological vector spaces and on nuclear spaces, and he made a fundamental contribution to functional analysis and what is extraordinary, it is that one can think that what made Grothendieck branch off, and which eventually led him to the idea of the topos, to this wonderful idea, this is his second thesis. Why ? Because the Grothendieck’s second thesis is written in this text, it is on the sheaves theory.
And then on this page, if you are perceptive, you will find that there is an error, which shows that you are never safe from mistakes. Because there is one of the examiners, if you look closely, who is called Georges Choquet(laughs). So I looked, to say maybe I was wrong, there are 3 examiners, there is Henri Cartan, there is Laurent Schwartz and then there is Georges Choquet. So I have searched on wikipedia to see if there was not a mathematician named Georges Choquet. In fact, no, there is a clergyman called Georges Choquet who died during the second world War. So there is no problem, it is a mistake, and it is Gustave Choquet who was Grothendieck’s examiner. So he sustained his thesis in 53.
And already in 55, he was of course interested in sheaves, which was a wonderful discovery of Leray.
And so there, I will start with some exchanges of letters between Serre and Grothendieck because finally, it is in these exchanges of letters that we see appear what is a so famous article that we call it The Tohoku.
This article appeared in a newspaper called Tohoku Maths Newspaper but the article is so famous that in fact it is called Tohoku.
So that’s what Grothendieck says ; he says :
“My Dear Serre,
Thank you for the various papers that you generously sent me, as well as for your letter. Nothing new for my part.”
Lecture by Alain Connes, November 7, 2017, as part of the ENS seminar “Grothendieckian Lectures” that can be followed here http://savoirs.ens.fr/expose.php?id=3257.
So that justifies what I wrote when I announced my presentation :
“I finished my annoying writing of homological algebra.”
So we will see very gradually what is the philosophy that Grothendieck uses all the time when he works, that is to say that he never hesitates before a task than any normal mathematician would consider it to be uninteresting, off-putting, not going to bring him anything. So he continues :
“... that I sent to Delsarthe who lacked the editorial staff for the typist.”
He says :
“I proposed it to Tannaka for Tohoku .”
Tohoku is Tohoku Maths Journal.
“It seems that the river articles do not put them off.”
So it is true that Grothendieck, in general, when he writes, the minimum is at least 100 pages. Then he talks about Weil. I’ll spare you what he says because he says :
“I have read at least the statements in Weil’s books on abelian varieties in the hope that we have arranged since the really discouraging demonstrations at Weil, his language disgusts me.”...
More !(Laughs)I pass... He says :
“I spend my time either learning or writing the varieties. It’s fun but long sure, but there is no question of research until you have swallowed a mountain of new things.”
So then, very important, this is probably the most important thing in this excha nge, he says to a given time :
“I realized that by formulating the theory of derived functors for more general categories than mo- dules...”
(it must be said that at the time, there was the book by Cartan-Eilenberg which was in the making, Serre calls it the Cartan-Sammy - because it’s Sammy Eilenberg - and in this book, there were of course the derived functors but it was still applied to module categories. That is to say, we took the category of modules on a not necessarily commutative ring and all the homological algebra was developed like that.
But obviously, it was very analogous in its formulation with what was happening for cohomology with coefficients in a sheaf. So what Grothendieck says is :
“I realized that by formulating the theory of derived functors for more general categories as modules, we get the cohomology of spaces with coefficients in a sheaf at low cost.”
You should know that at the time, when we took cohomology with coefficients in a sheaf, it was always Čech cohomology. That is to say that we took covers of the topological space, and then we made a complex or a bi-complex with these covers and we defined cohomology like that. Here.
So that’s what he says, and that point in his correspondence is absolutely essential. So then he continues, and so this is a letter from June 4, 1955, so we are 2 years after his thesis, so :
“Attached is the result of my first cogitations in form, on the foundations of homology.”
So afterwards, I won’t detail the rest since it’s on spectral sequences, etc. But let’s say that there, Gro- thendieck explains that he had planted himself on the existence of enough projective sheaves but at that time, he demonstrated that there were enough injection sheaves, and that allowed him to define, if you want, cohomology with coefficients in a sheaf without assumption on topological space. If we have good hypotheses on topological space, at that time, it coincides with the cohomology of Čech. But this is not generally true. So here is the Serre’s response. So there was something else in the correspondence, there
was something else that was what is called Im1, i.e. the functor for projective limits, how it commutes with the cohomology. But what is to read is the second paragraph.
“The fact that the cohomology of a sheaf is a special case of derived functors, at least in the para- compact case...”
(because in the para-compact case, it coincides with the cohomology of Čech, so the one which is defined from coverings)
“... is not in Cartan-Sammy.”
(Cartan-Sammy, this is Cartan-Eilenberg)
“Cartan was aware of this.”
So Cartan was aware that when they had developed the whole cohomological theory on the modules with Eilenberg, he was of course aware of the analogy with the case of the cohomology of sheaves, but hey, they didn’t want to bother themselves to do it in their book, and Cartan had conscience and had told Buchsbaum to take care of it. So it’s actually Buchsbaum who, independently of Grothendieck, also defined the abelian categories. He had started to develop it but he hadn’t been involved in cohomology.
“But it doesn’t seem to me that this one did. So the point of this would be to see what are just the properties of the fine sheaves to be used. So maybe we could realize if yes or no...”
(it’s Serre speaking, of course)
“... there are enough fine sheaves in the non-separated case.”.
So the non-separated case is extremely important of course for algebraic geometry and it was a time when Serre was developing algebraic geometry from Leray’s sheaf theory taking Zariski topology and Za- riski topology, at the start, it is not separate. I mean, so, we have exactly that problem. So this exchange is extremely important, it’s an exchange dating from the year 55. And Grothendieck’s article therefore, it is markedReceived 1 st March 1957, so it is this article“On some points of homological algebra”which is really the ancestor, we can really place, if you want, the origin of the topos in this article.
The reason we can place the origin of topos in this article is that in this article, if you want, first, of course, he introduces what abelian categories are. So this is extremely important, with all their properties, etc. He develops homological algebra in the context of abelian categories. So that’s what he does, if you want. But, what is much more important, is that he takes two types of examples, in his article, of abelian categories. The first example of abelian category that he takes, it is the abelian category of modules on a ring. It belongs to Cartan-Eilenberg. There is no problem there, but he also takes the example of the sheaves of abelian groups on a topological space, of course ; again, no surprise since it was for unify the two that he had done his generalization work. But what is absolutely crucial is that he had another example in mind, a third example in mind, and that’s what he called the categories of diagrams. That is to say, what he was doing was that he had the idea that if you take diagrams of abelian groups, but whatever diagrams you look at, well, that still forms an abelian category. Well actually, if you think right, you realize that he had the two pillars of the concept of topos. That is to say, he had the notion of topological space, which gives the notion of sheaves on abelian groups, etc., and he also had the notion of categories of diagrams and we will see that these categories there, they give rise to a topos. And these topos have an absolutely fundamental role that we will use it all the time, all the time, all the time.
So there is one thing that should be noted : he defines what an abelian category is. So what Grothen- dieck says is that an abelian category is an additive category. So I will explain to you in two words the misconception that we have on the additive category which satisfies the two additional axioms following : then, there is the fact that a morphism must have a nucleus and a co-nucleus, that, it’s an abstract notion, if you don’t know it, well, it would take a while, it’s a bit of gymnastics, and the second condition is being an exact morphism. That is to say, the fact that if you divide by the nucleus and if you look at what is called the image by defining it with respect to the co-nucleus, well then you have an isomorphism between the quotient by the kernel and image. And that is extremely important, it is not true for applications in
topological spaces for example, if you take an Hilbert space and if you take the continuous linear appli- cations in Hilbert space, that does not satisfy these two conditions. It is not an abelian category and the reason is that you can have a morphism which has an image, but it is dense in its image and its image is not closed and at this time, the second condition AB2 will not take place.
So there is a misconception that Grothendieck reproduces when he defines the additive categories in his article and that is the following : that in general, people define an additive category by saying “an additive category, it is a category where we add an additional structure which is the additive group struc- ture on morphisms.” Well, this is an heresy. I will explain to you why : because in fact it is not at all an additional structure. And there is an exercise in MacLane that I noticed for you there which shows that it is an heresy. What is this heresy ? The heresy is that when you take a category like an abelian category, it has products and co-products. This is a very simple thing. And it has an object which is both initial and final, it is object 0 ; in the abelian groups, the group is reduced to 0. Okay ? It is an initial object because you have a single arrow going from 0 to any abelian group, and it’s a final object because you have a single arrow going from an abelian group to 0. Everything is sent to 0. So there is an object that is both initial and final. Well, if you have that, you can tell, when the category is abelian. How ? Well, because what happens is that you have a completely canonical arrow, completely natural, which goes from the co-product of two objects to the product of two objects. Because using the 0, you can send half on 0 and the other half on 0 and at that time you have an arrow which is completely natural. If you asked if this arrow is an isomorphism, you did half the work. Because as explained in this MacLane text, at that time, you have an addition for morphisms. Simply by using the fact that this natural arrow which goes from the co-product towards the product is an isomorphism. You have a natural addition for morphisms and once you have this natural addition, you can take the additional axiom that there is a minus sign, if you want, for this addition, of course in general, you will not have a minus sign, that’s what we call semi-additive categories, they are very interesting, but if you want a category to be additive, you just ask that there be a reverse. So then, it’s not an additional structure. It is an heresy to believe that an additive category is given by a category + an additional structure. It’s not true. So it’s a misconception.
So, now I’m going to read Grothendieck, since the principle of the seminar is to fade away in front of Grothendieck. And even at some point, we will listen to him. And then when we have read and listened enough Grothendieck, there I will take a metaphor, then we will see an example. Okay so be patient, I’m not just going to read or listen to Grothendieck, you have to be patient, but let’s listen to him anyway.
Here is what he says :
“The point of view and the language of the sheaves introduced by Leray led us to look at these “spa- ces” and “varieties” of all kinds in a new light. They did not touch, however, the same notion of space...”
So what Grothendieck says is that Leray had envisioned a space in the form of sheaves on this space, but in fact, moreover, Leray only considered the sheaves of abelian groups. And we will see the change that Grothendieck has already made even there.
“They did not touch, however, on the very notion of space, only making us appreciate and more finely, with new eyes, these traditional “spaces”, already familiar to all. But it turned out that this notion of space is inadequate to account for “topological invariants” essentials which express the “form” of “abs- tract” algebraic varieties (like those to which Weil’s conjectures apply ), even that of general “schemes”...”
So here is the moment when I will make you hear Grothendieck. Why, because we are going to hear Grothendieck who defines what a sheaf is. So, I think it’s important that you listen to this, okay, because you don’t necessarily know what a sheaf is, we’re going to listen Grothendieck talking about this, okay, and once we’ve heard Grothendieck talking about it, we’ll come back to our sheep. This is the start of his conferences in Buffalo.
“Topoi were objects for years essentially on general topology. I mean a topos could be considered as the main object of study of topology. And so topoi is generalization of classical general topology. It’s what I really like to consider... Its study require some familiarity with handling topological spaces and continuous maps, and homomorphisms, and such things, and on the other hand familiarity with the language of ca- tegories... Later we... exponentiations... and give examples. But in order to understand... what a topos means..., one would require some familiarity with the language of sheaves on a topological space. I guess
that this notions are not very familiar for everybody, therefore I think I would give rather some introduction to sheaves on a topological space. I want to assume anything known on this matters. I would give a review of standard “sheafsury”... I hope that if some explanations are not clear, or if some comments come up that you’ll freely interrupt me to ask questions, or to point out errors or to make any kind of comments.
It would be nice if, as time goes on, there would be some kind of participation of the audience, I am sure that a number of you know something about topoi.... You will be able to make suggestions and comments.
So I start with a kind of formal survey of sheaves on a topological space. Provided an X to be a topological space, and consider the setO(X), it depends onX, of all open subsets ofX. The topology ofX is defined in terms of the family of open subsets ofX which is a subset ofP artiesdeX. I recall the axioms of the topology is thatOshould be stable under arbitrary unions and under finite intersections. All right. So O(X)in particular is an ordered set by inclusion and that, that all defines a category by abuse of language.
... any all upset does. If U andV are elements of O(X), are open sets on X, the set of homo- morphisms of U into V will be either empty if U is not contained in V and may be reduced to just the inclusion map ofU intoV, ifU is contained inV ; this is the definition of empty and of the decomposition of arrows... There must be at most one arrow from an object to another. So this construction of a category in terms of open sets makes sense for any all upset whatever. So the category which behave as... arrows of the graph of all the relations... Now let’s first define pre-sheaves : a pre-sheaf onX, sayf, is by definition a functor which goes from the category O(X) to the category of sets, when I say a pre-sheaf, I mean a pre-sheaf of sets, later we will see other types of pre-sheaves. But it should a contravariant functor... It’s a functor which goes from the opposite category to O(x) to the category of sets, so let’s recall what this means for a functor : first of all, it means for the objects of that means for every open set, U(X), we associate an object f U of the category of sets..., Remarque d’Alain Connes : ce qui est très important, c’est qu’il parle de faisceaux d’ensembles, et Leray parlait de faisceaux de groupes abéliens.It is an arrow of categories, that means that for every inclusion mapU toV two open sets, we associate a map fromf V intof U..., this map will be called the restriction map, corresponding to the pref-sheaf, and the axioms are the evident axioms of transitivity, namely if we have V is contained in another open setW, then we have a restriction map from f V to f W but also from f W tof V and we want the restriction fromf W tof U would be the composition here and moreover we want that in the case than we take the identity arrow, the identiry arrow from U into U, you want that the corresponding map f U goes tof U would be identity. ...
as maps of functors... So a pre-sheaf onf is just a contravariant functor from the categoryO(X)to sets is justO to sets. And the category of pre-sheaves on X, let us recall it P reSheaf(X), is defined as being the category of all functors fromO˙ to sets. So the pre-sheaves can be viewed as being objects of a category, the category of pre-sheaves, in the category of functors.
So what is an homomorphism of a pre-sheaf F into another G, by definition of homomorphisms of functors, let’s say f be such an homomorphism by definition, f consists in a connection of maps from f U intogU, we call this map f(U)for every U object of the category that of all open subsets ofX, and these maps of sets will be compatible with restriction maps, that means that whenever U is contained in an open set V, then we have alsof V that goes into gV by f ofV and we have the restriction maps here, from f V tof U and from gV, togU and so that the square should commute. So that’s an homomophism of pre-sheaves, that is just an homomorphism of functors, and they compose in an evident way, their composing is out and ? ? ?. There is an homomorphism from the pre-sheaf... that means for every U, an homomorphism from gU into hU, the composant of g and f is defined as associating for every U the composition of homomorphisms here.
All right, so that’s just general nonsense1on functors, on categories. Now, so far, we have not used the fact thatO(X) was the category of open subsets ofX, we have just used the fact that it’s an ordered set, but we are going to use it now in order to define the notion of pre-sheaf on X which recall sheaves onX, we have to introduce another axiom on pre-sheaves which will turn on to sheaves. Now, traditionnally the axioms on pre-sheaves is separate in two : you say first that the pre-sheaves are separators if the following is true : for every open setU on X and for every covering of U, open subsetsUi a covering is a union of the Uis, can be defined in terms of ordered sets ; it’s just the supremum of theUis, in terms of ... Be f a mapping of f U into each one of the f Uis, a restriction map, and therefore, we get a mapping of f into the product of thef Uis and have that two sets separated by every such choice here, this mapping is injective.
Now, let’s state this in another way : iff is a pre-sheaf onX,f ofU, the elements of the sets accord
1. ?
to sections of f over U. When we’ll give examples, we will see where this accord of sections come from.
All right, and, the map here I already said contained in V, the given map off V intof U will be called the restriction map... that mean the sections of ... is called the restriction of ... toU and the axiom for a sheaf to be separate means that whenever there is a union of open subsets Ui(X), from which the union is U, there’s a section of the pre-sheaves overU is known as we know these restrictions on theUis. The first condition would be the restricted in ... but in geometric terms, it just means that ... is an injective arrow, that means that a section off overU can be identified for the system of sections off over theseUis, and then the second question that arise is to see whether we can identify the subsets here that we obtain as image of f U which are the systems of sections of f over Uis which come from global sections f overU. Now let’s take a system of sections sayΦi inf Ui for everyi, for every index, here the necessary condition for this system ofP hiis to come from global sections which is the following : when you have two indicesi andj, the restriction of P hii toUi∩Uj could be equal to P hij...”
Here, I stop here, our patience is probably exhausted. But, I will say that one of the reasons for which I made you hear Grothendieck is quite complicated : we have to get used to this incredible patience he has to explain all the details, to come in, to go through all the details. And that, we will see, I mean, it is an absolutely essential quality in his approach. So I continue to read what he said about the point of view and the language of sheaves introduced by Leray. Grothendieck continues, he says :
“For the expected “marriage of the number and the size”, it was like a bed definitely narrow, where only one of the future spouses (that is, the bride) could at least find a place to nest somehow, but never both !”
So there, if you will, he actually had in mind all kinds of developments that were of combinatorial nature, which were related to number theory as opposed to what was going on in topology but hey, of course, there was Serre’s work on the use of Zariski’s topology.
“It is the point of view of the sheaves which has been the silent and sure guide, the effective key (and by no means secret), leading me without procrastination or detours towards the nuptial chamber with the vast marital bed. A bed so vast indeed (like a vast and peaceful very deep river...), that
“All the king’s horses could drink together...””
We’ll come back to that.
“As an old tune tells us”
(which I will make you hear later)
“As an old tune tells us that you must have been singing too, or at least hearing sung. And whoever was the first to sing it felt better the secret beauty and the peaceful force of the topos, that none of my learned students and friends of yesteryear...” (laughs)
Well, there is of course in this sentence, the fact that we know, without doubt, many of us, which is that the first to reveal a certain mathematical landscape has an apprehension of it which is incompa- rable (this is what happened to Galois for example) compared to the other mathematicians who come after him and understand him. This is a very striking thing, he says something more nasty, of much nastier.
“The key was the same, both in the initial and provisional approach (via the very convenient concept, but not intrinsic of the “site”)” (which I will tell you about, of course)“... than in that of the topos. This is the idea of the topos that I would now like to try to describe.”.
So let’s let Grothendieck speak for sure.
“Let us consider the set formed by all the sheaves on a (topological) space”
So, for the mathematician, frankly, these are sheaves of sets, that is absolutely fundamental, it is an enormous step, that it replaced the sheaves of abelian groups, which were interesting, we thought they were the only interesting ones, since they are the only ones who will give a cohomology, etc. No ! He had
the idea, which seems completely naive, to replace the sheaves of abelian groups by sheaves of sets, and we will see the range it gives.
“I believe elsewhere” (and that’s what he says)“to be the first to have worked systematically with sheaves from 1955.
What is the advantage of working with sheaves of sets, as we will see, is that when you work worth with the sheaves of sets, you can define what a group is in this stuff, you can define what an algebra is in this thing, because what you do is that you work as if you were working in sets but there is variability.
That is to say, there is something that moves, but otherwise you do exactly as if you were working in sets, as usual. So you can define what a group is. So if you’re looking for what an abelian group is in this category of sheaves of sets, well, you find the sheaves of abelian groups. So we fall back on our feet.
Then what Grothendieck says :
“We consider this “set” or “arsenal” as having its most obvious structure, the one that appears there, if one can say, “at first sight” ; namely, a so-called “category” structure.”
So of course, we were educated, at least in my time, with set theory. In fact, it was probably a mistake.
The real way of thinking is category theory. So he thinks of this kind of theory that he has before his eyes as a category.
“(Let the non-mathematician reader not be confused, not to know the technical meaning of this term. He won’t need it for the future.)”
That is to say, what the non-mathematician reader simply has to think is that he has an analogous, now, of set theory, in this new category, in this category of sheaves of sets. There is something that re- sembles set theory, and we will see that this metaphor goes very far.
“It’s this kind of “surveying superstructure”, called “sheaf category” (in space envisaged), which will henceforth be considered as “embodying” what is most essential to space.”
So we will see a metaphor, which I will develop further, but I can already reveal a part of it. If you want, usually, when we talk about a space, I’ll show it to you right away because I don’t want to wait, for this metaphor. Here, the metaphor is as follows : if you want, before Grothendieck, we used to, when we were studying a space, I don’t know, a curve, or wearing what, we put space... on the stage. And then we looked at it, we studied it, as a whole with structure, etc. Well, what Grothendieck does is... no ! Space is not on the stage, space, it’s behind the scenes. On the scene, there are the usual actors of set theory : the abelian groups, algebras, etc. But, the space in question, it is behind the scenes, like a species of Deus ex machina which introduces variability in the characters who are on the scene. That is to say that now the characters who are on the scene, they will depend on a hazard. This hazard is governed by the topos.
And when we have a Grothendieck topos, there are also the constants, that is to say that there are also the sets which do not depend on the hazard. And cohomology, it is defined by comparing the two.
So it’s absolutely fundamental that you gradually try to acquire a mental image, even if you are not a mathematician, to understand that the space which is given by the topos, it will appear behind, it is behind the scenes, it is not at the front of the stage, it is not it that we studied ; we study set theory, but there is this good god of topos, which is hidden and which makes everything vary, which introduces variability in there, okay. So then, that’s what Grothendieck says in another way : what he considers as
“this set or “arsenal” as having its structure the more obvious”, he thinks of it as a category, this category of sheaves of sets. So then what’s he saying ?“That it is a lawful thing to forget the space and to consider only the category of sheaves of sets.”
Why is it a legal thing ? It’s a legal thing because we haven’t lost space along the way. We find the points of space. So how do we find the space points in the metaphor I gave you ? Because he says to him
“it’s a simple exercise to check : once the question is asked.”. Well. You can actually have fun, thinking in classical terms, if you have the sheaves of sets on an ordinary topological space, how are you going to find the space itself, that is to say the points of space ? Can you ask this question. So in fact, in the metaphor I gave you, the points of space are when you take a given moment, a fixed time. When you take a time
that is frozen, well then there is no more variability, and you have ordinary set theory. That’s what we abstractly call in the theory of topos a point of a topos, that is to say that this is how we call a geometric morphism, which goes from ordinary set theory to the topos considered. But that is exactly like freezing things at a given time.
Grothendieck tells him“to check is a simple exercise.”. In fact, this is not always true ; as he says“(at intention of the mathematician) Strictly speaking this is only true for so-called “sober” spaces”. There still must be a minimum of separation in space. And there is an extremely interesting, which I invite you to do, because you always have to... we don’t do math by listening, we do math by doing exercises.
So there was already the exercise on the abelian categories of everything on time. And that’s another exercise now. Is that you take a curve with its Zariski topology. Or you rather take a surface with its Zariski topology. And you calculate the points of the corresponding topos, i.e. of the topology of the sheaves for the topology of Zariski. Well, you will notice that there are more points, because the space in question, it is not sober, and you will get exactly the points of the corresponding diagram. So I’m going to say, already, in this example, we see the absolutely incredible potential of this way of thinking. He says :
“... we can now “forget” the initial space, to no longer retain and use only the associated “category”, which will be considered the most adequate embodiment of the logical “topos structure” (or “spatial”) that needs to be expressed.”
So what he explains next is :
“As so often in mathematics, we have succeeded here (thanks to the crucial idea of “sheaf”, or “coho- mological meter”) to express a certain notion, (that of “space”), in terms of another (that of “category”).”
So we replaced, always following the metaphor, space by this category, which is a category type of sets, these are sets.
“Each time, the discovery of such a translation of a notion (expressing a certain type of situations) in terms of another (corresponding to another type of situation), enriches our understanding.”
Of course.
“And both, by the unexpected confluence of specific intuitions that relate either one or the other. Thus, a situation of a “topological” nature (embodied by a given space) is here translated by a situation of an
“algebraic” nature (embodied by a “category”) ; or, if you like, the “continuous” embodied by space, is
“translated” or “expressed” by the category structure, of an “algebraic” nature (and hitherto perceived as being essentially “discontinuous” or “discrete”).”
We deny, a priori, to a category the right to represent something continuous. This is the case here. This is the case because we find the points and we find the topology of the points, simply from the category, whichlooks likethe category of sets.
“But here there is more. The first of these notions, that of space, appeared to us as a notion in a
“maximum” way - a notion so general already, that one can hardly imagine how to find for it one more extension that remains “reasonable”. On the other hand, it turns out that on the other side of the mirror, these “categories” (thus the categories which one obtains as categories of sheaves of sets)“that which one finds, starting from topological spaces, are of a very particular nature..
The “mirror” in question here, as in Alice in Wonderland, is the one that gives as an “image” of a space, placed in front of it, the associated “category”.”
So this category if you want, it is the category of the scene behind which is the topos.
“They indeed have a set of strongly typed properties, (we will not talk about them all immediately) that make them look like sort of “pastiches” of the simplest imaginable of them.” What is the simplest of them ? It is set theory. So the categories you get like that, from a topos, are pastiches of set theory. That’s what Grothendieck says.
“That said, a “new style space” (or topos), generalizing traditional topological spaces, will be descri- bed simply as a “category” which, without necessarily coming from an ordinary space, has nevertheless all these good properties.” So they’re called topos, so he’s going to say it by the way, wait, I have to find him...
“The name “topos” was chosen (in combination with that of “topology”, or “topological”) to suggest that it is the “object par excellence” to which topological intuition applies. Through the rich cloud of images mentally aroused by this name, it must be considered to be more or less the equivalent of the term (topolo- gical) “space”, with simply a greater emphasis on the “topological” specificity of the concept. So...” Well, he’s talking about vector spaces, etc. So I go back.
“So here’s the new idea. Its appearance can be seen as a consequence of this observation, almost chil- dish to tell the truth, that what really matters in a topological space is nully its “points” or its subsets of points, and proximity relationships, etc. between these, but that these are the sheaves on this space, and the category they form. All I did was to lead towards its ultimate consequence the Leray’s initial idea - and this done, take the plunge. Like the very idea of sheaves (due to Leray), or that of the diagrams, like any “big idea” which shakes up an inveterate vision of things, that of topos has what to disconcert by its character of naturalness, of “evidence”, by its simplicity.”
In fact, we know when we do math, when we are on the right track, when someone tells you“Oh, that’s it !”.(laughs)
This is what Grothendieck says, so :
“... by this particular quality that makes us shout so often : “Oh, that’s it !”, in an half-disappointed, half-envious tone ; with in addition, perhaps, this implied “wacky”, “not serious”, that we reserve often to all that confuses by an excess of unforeseen simplicity. To what comes to remind us, perhaps, the long buried and denied days of our childhood...”
So there, he comes back to the concept of schema :
“It constitutes a vast extension of the concept of “algebraic variety”, and as such, it has renewed from top to bottom, the algebraic geometry bequeathed by my predecessors. That concept of topos constitutes an unsuspected extension, to put it better, a metamorphosis of the concept of space.”.
If you will, which is absolutely extraordinary, even at the outset, in the concept of topos, it is the way space is understood. As I said, it is no longer apprehended by points, it is apprehended by the hazard it introduces : it introduces a hazard into set theory ; he introduces a variability in set theory. And that is extraordinary.
“By this, it carries the promise of a similar renewal of the topology, and beyond this, of geometry.
From now on, moreover, it has played a crucial role in the development of new geometry.”
This is forl-adic cohomology, or for crystalline cohomology.
“Like her older sister (and quasi-twin)2, she has the two essential complementary characters for any fertile generalization, here it is.”
First, this notion should not be too broad, I pass quickly enough on it ; he talks about topos, we talked about it ;“the most essential geometric constructions must of course apply, that they can be transposed in a more or less obvious way.”. It shouldn’t be too general, the notion you take should not be, for example, the general notion of category. If it was the general notion of category, we would not go far. So it must have this property.
He explains : “Among these “constructions” , there is in particular that of all familiar “topological invariants”.”
He explains very well :“For the latter, I had done everything necessary in the article already quoted from “Tohoku” (1955).”
2. schema theory
So the origin, you see, it comes from there. As I said earlier, in Tohoku’s article, there were both the sheaves on topological space and there was also, and it was extremely important - so much important, the categories of diagrams, he was talking about the categories of diagrams and we will see that they play an absolutely essential role, in the same way. So he talks about mental associations, and less technical notions, of course.
“Second, the new concept is at the same time broad enough to encompass a host of situations which, hitherto, were not considered to give rise to intuitions of a “topologico-geometric” nature - to the intui- tions, precisely, that we had reserved in the past only for ordinary topological spaces.”
As we will see in an example that I will give you in a relatively short time, as we will see, what hap- pens in the metaphor I was talking about, what will happen is that as it there is this hazard, as there is this variability in set theory, we can no longer apply the excluded middle principle. On the other hand, intuitionism works. And so, what it will generate, this nuance, that’s why I want to take you there step by step, we’re slow, but we have to be slow, so I will show you an example, as I said at the beginning, in which the notion of truth associated with topos will be much more subtle than the notion of ordinary truth, and I will have a slide on which there will be marked “a step from the truth” and I will give you a topos in which we will be “10 steps from the truth”, “15 steps from the truth”, “20 steps from the truth”, etc. We will take an example because until we have taken an example, as long as we talk abstractly, we don’t really know what we’re doing. So we’re headed there.
“The crucial thing here, from the perspective of Weil’s conjectures, is that the new notion is pretty vast indeed, to allow us to associate with any “diagram” such a “generalized space” or “topos” (called “étale topos” to the proposed scheme). Certain “cohomological invariants” of this topos (all that there are “dumb kids” !) seemed to have a good chance of providing “what we needed””.
He continues and we relax a bit before coming to the examples and the really crucial things.
“It is in these pages that I am writing that, for the first time in my mathematician life, I take the leisure to evoke (if only to myself) all the master-themes and great guiding ideas in my mathematical work. This brings me to better appreciate the place and the scope of each of these themes, and of the “points of view”
they embody, in the great geometric vision that unites them and from which they come. It is through this work that the two innovative nerve center ideas appeared in full light, in the first and powerful development of new geometry : the idea of diagrams, and that of the topos.”
And there he insists :
“It is the second of these ideas, that of topos, which now seems to me to be the deepest of them. If by chance, towards the end of the fifties...”
So Grothendieck introduced the topos in a slightly depressed period he had after the death of his mother in 1957, he introduced the topos in 58. So we will be, in the coming year, in the 60th anniversary of the birth of the topos.
“If by adventure, towards the end of the fifties, I had not rolled up my sleeves, to develop stubbornly day after day,”
This is Grothendieck : “Obstinately, day after day...”
“Throughout twelve long years, a “schematic tool” of delicacy and power perfect - it would seem almost unthinkable to me that in the ten or twenty years already that have followed, others that I in the long run could have prevented from introducing at the end of the ends (were it to their bodies defending) the notion which obviously imposed itself, and to draw up somehow at least a few dilapidated “prefab” barracks, failing the spacious and comfortable3 residences that I had at heart to assemble stone by stone and go up with my hands.”
3. address of the speaker to the audience : “there, he is talking about the diagrams”
There he talks about schemas.
“On the other hand, I don’t see anyone else on the mathematical scene, over the past three decades, who could have had this naivety, or this innocence, to take (in my place) this other crucial step between all of them, introducing the so childish idea of topos (or even that of “sites”). And even assuming that idea already graciously provided, and with it the timid promise it seemed to harbor...”
You know, someone would tell you, “I’m going to do this...”. “Good luck !”, you would say ! Okay...
(laughs4)
“... I don’t see anyone else, either among my friends of yesteryear or among my students, who would have the breath, and above all faith, to bring this humble idea to fruition (apparently ridiculous...) “What is it to strive on the sheaves of sets on a topological space ?”... (seemingly derisory when the goal seemed infinitely distant...) : since its earliest early beginnings, until the full maturity of the “mastery of eternal cohomology”, in which it ended by incarnating in my hands, in the years that followed.”
Good, after, he talks about details, finally, of things which are important for the mathematician, but he speaks of etale cohomology and it is on this subject, as he says :
“It was inspired by this statement that I discovered the concept of a site in 1958.” So he discovered the notion of site in 1958, and it is this notion, of course, and cohomological formalism, which were developed later.
He says :
“When I speak of “breath” and “faith” ,” (that’s always for the mathematician), “these are qualities of “non-technical” nature.”
Grothendieck wrote somewhere inCrops and Sowing that he was not fast, that he was surrounded by people much faster than him, but hey, this is an example that shows how not to be discouraged, when you are not fast, good when talking with people who you realize they understand ten times faster than you do, don’t be discouraged. However, which is absolutely crucial is to be persistent, and to have faith in an idea.
That is to say, if you have an idea, you must first make it your own, make it your own. And once it’s yours, you have to protect it ; initially, you have to protect it like a very small child who has just been born. Do not show it too much, not too much, etc.(little laughs). And then after... Not so much because someone can take it from you but because you have to test it, we will talk more about it late, you have to test it, you have to get used to it, in private.
“On another level, I could also add to it what I would call “cohomological flair”, that is to say the kind of flair that had developed in me for building cohomological theories.”
Afterwards, he groans a little against his pupils, but that, we are used to with Grothendieck.
We’re going to take a short break : we’re not going to stop but I’m going to make you listen to Yves Montand(laughs).
“Yes, the river is deep, and vast and peaceful are the waters of my childhood, in a kingdom that I thought I quit a long time ago. All the king’s horses could drink there together at ease and all their drunk, without exhausting them ! They come from glaciers, fiery like these distant snows, and they have the soft- ness of the plains clay. I just talked about one of these horses, that a child had brought to drink and who drank his content, at length. And I saw another one coming to drink a moment, in the footsteps of the same kid if it is - but there it did not drag. Someone must have chased it away. And that’s all, as much to say.”
We hear to the song Aux marches du Palais who speaks about The girl who has so many lovers that she doesn’t 4. to underline the irony of the euphemism of Good Luck compared to the magnitude of the task that this represents.
know which one to take and who has chosen her little shoemaker.
Here we go back to serious things.
So he goes on, he says :
“However, I see countless herds of thirsty horses roaming the plains - and not later that this morning their whinnies pulled me out of bed at an undue hour, I am going to my sixties and I love tranquility.
There was nothing to do, I had to get up. It makes me pain to see them, in the state of lanky rosses, while good water however is not lacking, nor green pastures. But it looks like a malicious spell has been cast on this land that I had known welcoming, and condemned access to these generous waters. Or maybe it is a coup mounted by the country horse traders, to bring prices down, who knows ? Or maybe it’s a country where there are no more children to lead the horses to drink, and where the horses are thirsty, for lack of a kid who finds the path that leads at the river...”
So, with Pierre Cartier and Olivia Caramello, we organized a conference two years ago, in the IHES, precisely to revive the idea of topos, but really of Grothendieck’s topos, and the colloquium was remar- kable, it went very very well.
So what is a Grothendieck topos ? So now we come back to mathematics. So there are three ways of defining them : it is undoubtedly the first way which is the simplest. So, if you will, as Grothendieck explains when he lectures, the important thing, at the start, they were pre-sheaves, that is to say they were contravariant functors which went from the open category, but it’s an extremely simple category. I mean it’s a category for which between two objects, there is at most a morphism so it’s really something extremely simple. So we looked at the contravariant functors that went from this category to the sets. So now, we remove all the conditions on this category, except that it is a small category. What does a small category mean ? It means that the objects form a whole. And then morphisms as well of course. So we look at a small category and we look at all the contravariant functors from this small category to sets. We forget the fact that we had them open and that we had an extremely special category by looking at the open. Okay, we take no wear which. And so now, what we’re asking, of course, we’re not going to take all the contravariant functors, since we know well, and Grothendieck explained it in what we listened to, that we are not going to take all the pre-sheaves. Among these pre-sheaves, we will select some which we will call sheaves. What is the important property of this selection ? There are two things that are very important in this selection : the first is that we are not going to change the morphisms ; the first thing that is fundamental is that when you take a morphism from one sheaf to another, you can forget that these are sheaves. It’s a pre-sheaf morphism, okay. So in fact, when we are going to select the subcategory of the pre-sheaf category, we will take a full subcategory. Full subcategory, that means that we are not going to change the notion of morphism. Okay, that’s crucial, that, if you listen to Grothendieck further, he talks about it and he says it is crucial. So first thing. Second thing, which is extremely important, is that there is a way, when you have a pre-sheaf, to harness it, to turn it into a sheaf. So that means that sheaves are special pre-sheaves, but there is a kind of projection that allows you to replace a pre-sheaf by a sheaf. So what is the correct way to say it is that the functor which includes the category of sheaves in the category of pre-sheaves, first it is full, it is faithful, because we forget nothing, but above all, it has an adjoint on the left, who is the harnessing, and miraculously, this adjoint on the left, it is exact on the left, which is nor- mally never the case for an adjoint on the left. Normally, an adjoint on the left, we know that in all cases, it will preserve those which are called colimits, but it is very rare that it preserves the limits. Well there, it preserves the limits. So that’s the condition. So if you want a short definition of what a topos is, that’s it.
So now what we will see, and then we will see what a site is, there is another way to say it which is more precise : it is that in fact, we know that any harnessing, like the one of which I was speaking, in fact, it comes from what is called a Grothendieck topology on the small category which we left. So let’s see what it is. And then in fact, there is a third definition of what is exactly a topos. But then, that is really a definition, how to say, very abstract but it is a definition that states properties that are true for set theory, okay. And we ask the topos to check them too. So that spawned another theory of topos, called the elementary topos theory, which are not Grothendieck toposes in general. But then, what is it missing to an elementary topos, therefore a topos which verifies naive properties of set theory, to be a Grothendieck topos ? What it lacks are the constants. That is to say that when, in the metaphor, I told you that we have variable sets, which depend on a hazard, well, when we have a Grothendieck topos, there is what is called a geometric morphism, which goes from the topos to the topos of the sets, and that allows us
to talk about the constants. Now to talk about the constants, when doing cohomology for example, it’s absolutely essential. Because these are the constants which allow for example to define the global sections of a beam, etc., etc. So it’s not at all innocent, and there is a very big difference between a Grothendieck to- pos and what is called an elementary topos which would bring together elementary properties of set theory.
So the examples. Well then among the examples, there is of course the example of sheaves of sets on a topological space. This is the first example. The second example, I already talked about it, are sheaves for Zariski topology, so this is a special case of sheaves of sets on a topological space. But as I said, the point is that when we look for the points, for this topos, we find the good points of the diagram, okay.
And finally, there is a third example, which is the one that Grothendieck introduced in 58 to have etale cohomology, that is to say that we start from a diagram, and there is a topos which is associated with the diagram, but it is no longer a topos which comes from a topology on the diagram. So this is something that is above and which, well, of course, there is already a topos in the sense original if you want, which is outside the topological spaces.
So I come back to Grothendieck, he says :
“The theme of the topos comes from that of the diagrams, the same year in which the diagrams appea- red - but in extent it goes far beyond the mother theme. It is the theme of the topos, and not that of the diagrams, which is that “bed”, or this “deep river”, where geometry and algebra, topology and arithmetic come together, mathematical logic and category theory, the world of the continuous and that of structures
“discontinuous” or “discrete”. If the theme of diagrams is like the heart of new geometry, the theme of the topos is the envelope, or the abode. It’s what I have designed more broadly, to grasp with finesse, through the same language rich in geometric resonances, a common “essence” to situations of the most distant from each other, coming from such and such a region of the vast universe of mathematical things. This theme of the topos is very far from having known the fortune of that of schemas.”.
There is a kind of curse on topos. There is a curse that reigns, we may come back to it if we have time. So here is the metaphor. So the metaphor I was talking about earlier. It is absolutely necessary that you have a mental picture of what a topos is. So we used to, like I said, to put the space to be studied on the front of the stage. Grothendieck makes it play this role of Deus ex machina, who is not present, who stays behind the scenes. But what is important is to know that when you have a topos, you can do all the manipulations, you can talk about abelian groups, you can talk about algebras, etc., and if you work with a topos coming from a topological space, that would give you the sheaves of abelian groups, or the sheaves of algebras, etc., it’s great, it’s great to have this freedom of maneuver. So then, when one works in a topos, everything happens as if we were handling ordinary sets. So that’s what one needs to know. In fact, as soon as we have bundles on a space, we get into the habit of thinking about a bundle as in a variable vector space. But there, variability is the rightnotion of variability, because it sets up the sets. Except that we can no longer apply the rule of the excluded middle. So what appears if you want is that we can no longer have, for a proposition p, (pis true) or (not pis true), we no longer have the excluded middle principle. So we will quickly see a concrete example of a topos for which this notion of truth becomes more subtle than the simple true or false than we use colloquially. For example, if you want, if you watch TV, and you watch a political discussion on television ; well, we used to say “this one is right and this one wrongly”. Well, I pretend that we don’t have the conceptual tool we need to judge. And I will give you examples. I’m going to show you how much more subtle the notion of truth is and how the idea of the topos allows to formalize it. So we’re going to make this work on an example. To make that work, we are going to introduce topos which are other than the topos which come from a topological space and which have an extremely simple nature : these are the topos which consist in taking a small category and simply take the category of all contravariant functors to the sets. So here we does not distinguish between sheaves and pre-sheaves. We take all the pre-sheaves. They are all sheaves. So to a small category, we are going to associate a topos which is its kind of dual, if you want, which is all the contravariant functors from this little category to the sets, and we will have fun with that.
So in 91, Grothendieck was still in perfect contact with certain mathematicians, and lo and behold what he wrote in a letter to Thomasson, he said :
“On the other hand for me, the original paradise for topological algebra is by no means the simplicial category...” . So, I don’t know if we will have time to talk about it, but he is talking about topos.
Indeed, topos having categories sheaves on setsC, withˆ C a small category, are by far the simplest of the known topos.”And it is for having felt, that he insists so much on these categorical toposes in SGA4. So if you look at SGA4, you will see that there are two basic topos examples ; of course, there is the etale to- pos, and then there are the topos which are dual of a category. Okay. So we’re going to have fun with that.
So what is the notion of truth in a topos ? (laughs) How is the notion of truth different in a topos ? So, in what sense, first of all, are we able, in the sets, to define the true and false ? So how are we going to define true and false in set theory ? We are going to be interested in trying to classify the subsets of a set, okay. You see, if you work with ordinary sets, and if I tell you “I have a functor which, if you give me a set, it combines all of its subsets.” . It is a functor because if you have an application that goes fromX inY, you can recall the subsets of Y backwards, so it’s a functor. So now, the question is “is this functor representable ?”. It’s a mathematical notion, okay, and then in the sets, it is representable because of a notion that we know very very well : it is that with a subset, we associate what is called its characteristic function. That is to say that when we have a subset of a set, we define a function : this function, it is 1 if we are in the subset, and it is 0 if we are not in the subset. So this function, this makes that it has a fairly miraculous property : it classifies, that is to say, it represents this functor. In the case of sets, there is a privileged object Ω which is the object which is formed of the set{0,1}, the two-point set, and when you look at all the subsets of a set, it comes back to look at all the applications of this set towards the set{0,1}.
Because when you have a application which goes to the set{0,1}, the subset, it is defined by the subset on which it takes the value 1. But where it does not take the value 1, well, it necessarily takes the value 0.
Well, if we think enough, in logic, thinking in the language of topos, we realize that it is this simple fact that there were only 0 and 1 in the sets which makes it possible to have the principle of the excluded middle.
So now we’re going to have fun with a topos which is a little bit more complicated. We will take... this is what I call “at two steps from the truth”. So, what are we going to take as a topos ? We will take the categoryC which has only one object, and which has for morphism the powers of one morphism. That is to say that I choose a single morphism which goes from this object in itself and I raise it to powers, okay, Tn. So what does that mean, an object from the category of contravariant functors from this category to sets ? It simply means a set with an application fromX in X. That’s all. Why do you only have one set ? Because the category had only one object. So you only have one set. And in fact, the category, it had only one morphism, well, we raise it to its powers, but, I mean... you just have to know it, you just have to know its image. What is its image ? It is a transformationT fromX into X. What is the topos ? Well, these are the sets provided with a transformation. Well, the sets with a transformation, that makes a topos. It’s a category, okay. Why is it a category ? It’s a category because if you have two sets with a transformation, you have the applications fromX toY which respect transformation. That is, they verify that the image f(T X) ofT X isT f(x). So you have a category, and this category is a topos. Why is it a topos ? Because it is the dual of the small category that I gave you.
Okay so now we’re going to look for Ω, for that, so we will try to classify the sub-objects of an object.
So why is it annoying to try to classify the sub-objects of an object ? Well, let’s try with Ω ={0,1}. We will try with the characteristic function, as we did earlier. After all, if I take a set with an application, if I take a sub-object, it’s a subset that is stable by application, okay. So if I take myX, I will take a subset Y which was invariant, which was invariant by applyingT. Well. Very good. It is invariant by applyingT. So I’m going to associate the value 1 on this subset, okay. On the subset, I will give 1. Why can’t I give the value 0, on the complementary ? Well because there can have complementary points that will end up landing in the set in question. I’m not at all assured that the complementary will be invariant under T. It may very well happen that a point in the complementary, after a while, tac !, it will type in the subset in question. It’s not because the subset in question is invariant that its complementary is invariant. Of course no ! The transformation I took is an action onN, not onZ. So how do we go do ? It’s annoying ! It means that the application which went towards 0 and 1 does not work. Well ! Well, you have to think about it a little bit. What do we have to do ? Well, when I take an x which is in X, there will exist a smaller integer, a first integer, such as when I apply T ntimes, it hits the subset. Okay. So I’m going to associate this integer with it. This integer, it will be infinite, of course, if we never get into the subset.
So we see that we have to replace the set{0,1} by the set {0,1,2,3, . . . ,∞}. And how are making this set a set with a transformation ? Well, we realize that if I look theh, i.e. the smallest integer forT X, well the smallest integer for T X, it’s going to be the smallest integer for X minus 1 unless it becomes negative ; if it becomes negative, it doesn’t work ; so I take thesupwith 0. Okay.
So you see that for this topos, then the notion of truth which before was simply 0 or 1, main-holding, it is given by the set {0,1,2,3, . . . ,∞}, with the transformation which replaces byN−1. So what does that mean ? Well, that means we have an incredibly simple example of a topos which allows to say “Yeah, what you do, yeah, I would say it’s 10 steps from the truth...”. Me, I’ve always said that people who do string theory are infinitely close to the truth.(laughs)
So you see that this notion, innocent as it is, stupid as it looks, in fact, it has an absolutely rich potential.
And what I claim is that our mind, our training logic, is extremely primitive because we are used to, when we listen to a discussion policy of decreeing “yes or no”, “such a person is right, such a person is wrong”
and we are wrong by doing that and if there were philosophers, well, I dream, if there were philosophers knowing math, and who understand the topos from the inside, and there are very few, who understand the topos from inside, they would be able to give models, which would be useful, to appreciate much better these kinds of discussions, these kinds of situations, which are actually much more subtle compared to the notion of truth, that this notion of absolute ineffectiveness, that we use all the time, and that is “Such is right or so is wrong.”. So I absolutely wanted to give you this example, so that you keep it in mind, and try to build other similar examples ; there are examples of course, don’t be afraid that there arenthat goes from 0 to infinity, in fact, you can very well imagine finite constructions, okay. The finite constructions, there is a wealth combinatorial in the topos which makes finite constructions have extraordinary potential.
So what is a sieve ? This example will allow us to define what a sieve is. What is a sieve ? I gave you an example of a sieve. The Ω in general, when you take the topos, which is given by all the contravariant functors from a small category to sets, well we build the Ω and how is the Ω built ? The Ω, it is constructed from a sieve. So what is a sieve ? Well a sieve on an object of a category, the Ω will be constructed from the objects of this category, let’s remember that the category which I mentioned earlier, it had only one object.
So for the moment, we have nothing. We have a single object. So, a sieve on an objectX, on our object X, is the data of a family of morphismsC(X) which is contained in all the morphisms whose image is X... finally, it goes from a setZtoX, whose codomain isX, and which is stable by right-hand composition.
What are cribles, in the earlier example ? We had only one object ; the morphisms which went into this object, it was just the integers, since it was the powers ofT, there wasT0,T1, T2,. . .. What is a sieve ? Well, a sieve is a kind of ideal if you will, that is to say that it is a family of morphisms which is stable by right-hand composition by any what morphism. So in the case of earlier, what is the composition on the right ? It add to an integer, well, that adds any integer to it. It’s like looking at all the intervals infinite on one side. So, among the infinite intervals, you have what, you have 0, up to infinity, that is what is called... finally, it is a sieve which must always be present ; it is the sieve that is formed by all morphisms.
And then, we had all the morphisms that were from a certain integer n. That was a sieve, okay, and it was when we were at a distancen. And then, there is the sieve where there is none to none, it’s the empty set, and that corresponded to infinity earlier. Here.
So it turns out that in general, we can define the Ω, the truth values if you want, for the dual of a small category, and we define it exactly from the sieves. When we calculate Ω therefore, we construct this object, simply as always as a contravariant functor of a set etc., but we build it from the sieves on each of the objects in the category. In our case, there was a single object so it was very simple. It was very very simple.
So I was fascinated for a long time by the idea that Grothendieck had called sieve and that he was not unaware that this name had already been used by mathematicians, and that there is for example a sieve which is well known and which is the sieve of Eratosthenes. So I finally found the answer, I finally found why the sieve of Eratosthenes is a sieve, in the sense of Grothendieck and that, it comes from a common work we did with Katia Consani and in which the category we take is very similar to that from earlier, where there was only one transformation, but this time, it is a little more complicated anyway, because instead of having (we always have a single object, as before), but instead of having the powers of a single morphism, we have an action of the multiplicative integers. That is, for each integer, we have a morphism, and when we make the product of two integers, the morphisms are composed. So it is an exercise to demonstrate that the sieve of Eratosthenes is a sieve in the following way : it is very funny.
Because... what is the sieve of Eratosthenes ? The sieve of Eratosthenes, that is to take the first non-trivial number. We’re going to fuck 1, we don’t care 1, okay. So we take the first non trivial number which is 2.
And what does the sieve do ? The sieve considers all multiples of 2, all even numbers except 2. And then there are things, good. There are 3 left for example, then it takes all the multiples of 3 except 3. And then there are things, 4 we have already taken since... So it takes all the multiples of 5 except 5. Well, I
